13 problems

Let $E\to(X,\omega)$ be a Hermitian vector bundle, and let $h$ be a Hermitian metric on $E$. A strong subsolution for the $Z$-critical equation is a metric satisfying … as an…

Collins–Jacob–Yau hypercritical-phase conjecture. The set $\mathcal{H}_{\Omega}$ is non-empty and $\Omega$ has hypercritical phase if and only if, for every irreducible subvariety…

Let $(X,\alpha^{\mathbb{C}})$ be a complexified Kähler manifold in the hypercritical range, and let $\mathcal{H}$ denote the space of admissible potentials. C^{1,1} geodesic conjec…

Let $(X,\omega)$ be a compact Kähler manifold of complex dimension $n$, let $\alpha^{\mathbb{C}}=\alpha+\mathrm{i}\beta$ be a complexified Kähler class, and let $\hat{\vartheta}$ b…

Let $X$ be the compact Kähler manifold under consideration, and let $L$ be a holomorphic line bundle on $X$. A metric solving the deformed Hermitian–Yang–Mills equation is a metric…

Let $X$ be a rational homogeneous variety of complex dimension $n$, let $[\psi]$ be a class with phase angle $\hat{\Theta}\in \big((n-2)\frac{\pi}{2},n\frac{\pi}{2}\big)$, and for…

Collins–Yau's conjecture. The following are equivalent:

Let $(X,\omega)$ be a compact Kähler manifold of complex dimension $n$, let $[\alpha]\in H^{1,1}(X,\mathbb R)$ be a real cohomology class, and let $\hat\theta$ be a supercritical p…

Collins-J-Yau conjecture. The class $[\alpha]$ admits a solution to the deformed Hermitian-Yang-Mills equation with supercritical phase if and only if $Z(X)\neq 0$ and, for every a…

Let $(X,\omega)$ be a Kähler threefold and let $L\rightarrow X$ be a holomorphic line bundle. Suppose that the Chern number inequality holds, the lifted angle is well-defined with…

Let $X$ be a Calabi–Yau manifold, let $L\rightarrow X$ be a holomorphic line bundle, and let $L\hookrightarrow X$ be a Lagrangian submanifold. The objects $L$ are viewed respective…

Let $(X,\omega)$ be a Kähler manifold, let $L$ be a line bundle or let $[\alpha]$ be a real $(1,1)$ class, and let $Z_V(L)$ denote the central charge associated to each analytic su…

Let $(X,\omega)$ be a Kähler $3$-fold and let $L\to X$ be a holomorphic line bundle. Assume that $L$ admits a solution of the deformed Hermitian–Yang–Mills equation with lifted ang…